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MacOS 8.1

Win98

DOS

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|00005e70| 20 53 75 70 70 6f 73 65 | 20 24 66 27 5c 6e 65 30 | Suppose| $f'\ne0|
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|00005e90| 20 6f 66 20 74 68 65 0d | 64 6f 6d 61 69 6e 2e 20 | of the.|domain. |
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|00005eb0| 65 20 77 61 76 65 20 66 | 72 6f 6e 74 20 24 5c 66 |e wave f|ront $\f|
|00005ec0| 5f 74 24 20 66 6f 72 6d | 20 61 74 20 74 68 6f 73 |_t$ form| at thos|
|00005ed0| 65 0d 70 6c 61 63 65 73 | 20 77 68 69 63 68 20 61 |e.places| which a|
|00005ee0| 72 65 20 74 68 65 20 69 | 6e 74 65 72 73 65 63 74 |re the i|ntersect|
|00005ef0| 69 6f 6e 20 6f 66 20 74 | 68 65 20 65 6e 76 65 6c |ion of t|he envel|
|00005f00| 6f 70 65 20 66 6f 72 20 | 24 66 24 20 77 69 74 68 |ope for |$f$ with|
|00005f10| 0d 6d 61 70 70 69 6e 67 | 20 6c 69 6e 65 73 20 24 |.mapping| lines $|
|00005f20| 5c 4c 5f 73 24 2c 20 66 | 6f 72 20 74 68 6f 73 65 |\L_s$, f|or those|
|00005f30| 20 24 73 24 20 68 61 76 | 69 6e 67 20 24 66 27 27 | $s$ hav|ing $f''|
|00005f40| 28 73 29 20 3d 20 30 24 | 2e 0d 5c 6d 65 64 73 6b |(s) = 0$|..\medsk|
|00005f50| 69 70 0d 0d 7b 5c 62 66 | 20 50 72 6f 6f 66 7d 20 |ip..{\bf| Proof} |
|00005f60| 20 53 69 6e 63 65 20 74 | 68 65 20 56 69 73 75 61 | Since t|he Visua|
|00005f70| 6c 69 7a 65 72 20 6f 6e | 6c 79 20 64 69 73 70 6c |lizer on|ly displ|
|00005f80| 61 79 73 20 74 68 65 20 | 65 6e 76 65 6c 6f 70 65 |ays the |envelope|
|00005f90| 0d 66 6f 72 20 64 65 63 | 72 65 61 73 69 6e 67 20 |.for dec|reasing |
|00005fa0| 66 75 6e 63 74 69 6f 6e | 73 2c 20 77 65 20 77 69 |function|s, we wi|
|00005fb0| 6c 6c 20 61 73 73 75 6d | 65 20 24 66 27 28 73 29 |ll assum|e $f'(s)|
|00005fc0| 20 3c 20 30 24 2c 20 73 | 6f 20 74 68 61 74 0d 24 | < 0$, s|o that.$|
|00005fd0| 5c 7c 66 27 28 73 29 5c | 7c 20 3d 20 2d 66 27 28 |\|f'(s)\|| = -f'(|
|00005fe0| 73 29 24 2e 20 41 20 6e | 65 61 72 6c 79 20 69 64 |s)$. A n|early id|
|00005ff0| 65 6e 74 69 63 61 6c 20 | 70 72 6f 6f 66 20 77 6f |entical |proof wo|
|00006000| 72 6b 73 20 66 6f 72 0d | 70 6c 61 63 65 73 20 77 |rks for.|places w|
|00006010| 68 65 72 65 20 24 66 27 | 28 73 29 20 3e 20 30 24 |here $f'|(s) > 0$|
|00006020| 2e 20 53 69 6e 63 65 20 | 63 75 73 70 73 20 6f 63 |. Since |cusps oc|
|00006030| 63 75 72 20 61 74 20 70 | 6c 61 63 65 73 20 77 68 |cur at p|laces wh|
|00006040| 65 72 65 0d 24 5c 66 5f | 74 27 28 73 29 3d 5c 76 |ere.$\f_|t'(s)=\v|
|00006050| 65 63 30 24 2c 20 77 65 | 20 63 61 6c 63 75 6c 61 |ec0$, we| calcula|
|00006060| 74 65 20 74 68 69 73 20 | 74 61 6e 67 65 6e 74 20 |te this |tangent |
|00006070| 76 65 63 74 6f 72 3a 0d | 24 24 5c 66 5f 74 27 28 |vector:.|$$\f_t'(|
|00006080| 73 29 20 3d 20 5c 42 69 | 67 6c 28 74 76 5f 73 27 |s) = \Bi|gl(tv_s'|
|00006090| 2c 5c 20 31 2b 5c 62 69 | 67 6c 28 66 27 28 73 29 |,\ 1+\bi|gl(f'(s)|
|000060a0| 2d 31 5c 62 69 67 72 29 | 76 5f 73 74 20 2b 0d 5c |-1\bigr)|v_st +.\|
|000060b0| 62 69 67 6c 28 66 28 73 | 29 2d 73 5c 62 69 67 72 |bigl(f(s|)-s\bigr|
|000060c0| 29 76 5f 73 27 74 5c 42 | 69 67 72 29 24 24 0d 0d |)v_s't\B|igr)$$..|
|000060d0| 5c 6e 6f 69 6e 64 65 6e | 74 20 77 68 65 72 65 20 |\noinden|t where |
|000060e0| 24 76 5f 73 27 3d 20 2d | 4b 66 27 27 28 73 29 24 |$v_s'= -|Kf''(s)$|
|000060f0| 20 73 69 6e 63 65 20 24 | 5c 7c 66 27 28 73 29 5c | since $|\|f'(s)\|
|00006100| 7c 3d 2d 66 27 28 73 29 | 24 2e 0d 0d 54 68 65 20 ||=-f'(s)|$...The |
|00006110| 73 74 61 72 74 69 6e 67 | 20 63 75 72 76 65 20 24 |starting| curve $|
|00006120| 5c 66 5f 30 24 20 69 73 | 20 61 20 76 65 72 74 69 |\f_0$ is| a verti|
|00006130| 63 61 6c 20 6c 69 6e 65 | 2c 20 73 6f 20 68 61 73 |cal line|, so has|
|00006140| 20 6e 6f 20 63 75 73 70 | 73 2c 0d 68 65 6e 63 65 | no cusp|s,.hence|
|00006150| 20 77 65 20 6d 61 79 20 | 61 73 73 75 6d 65 20 24 | we may |assume $|
|00006160| 74 5c 6e 65 30 24 3b 20 | 74 68 65 20 6d 69 6e 69 |t\ne0$; |the mini|
|00006170| 6d 61 6c 20 76 65 6c 6f | 63 69 74 79 20 24 4b 24 |mal velo|city $K$|
|00006180| 20 69 73 20 61 6c 73 6f | 0d 6e 6f 6e 2d 7a 65 72 | is also|.non-zer|
|00006190| 6f 2c 20 73 6f 20 24 74 | 76 5f 73 27 3d 30 5c 69 |o, so $t|v_s'=0\i|
|000061a0| 66 66 20 76 5f 73 27 3d | 30 5c 69 66 66 20 2d 4b |ff v_s'=|0\iff -K|
|000061b0| 66 27 27 28 73 29 3d 30 | 5c 69 66 66 20 66 27 27 |f''(s)=0|\iff f''|
|000061c0| 28 73 29 3d 30 24 2e 0d | 54 68 69 73 20 62 65 69 |(s)=0$..|This bei|
|000061d0| 6e 67 20 74 68 65 20 63 | 61 73 65 2c 20 77 65 20 |ng the c|ase, we |
|000061e0| 73 65 65 20 74 68 61 74 | 0d 24 24 0d 5c 66 5f 74 |see that|.$$.\f_t|
|000061f0| 27 28 73 29 3d 5c 76 65 | 63 30 5c 69 66 66 20 66 |'(s)=\ve|c0\iff f|
|00006200| 27 27 28 73 29 3d 30 5c | 61 6e 64 20 74 3d 7b 31 |''(s)=0\|and t={1|
|00006210| 5c 6f 76 65 72 0d 76 5f | 73 5c 62 69 67 6c 28 31 |\over.v_|s\bigl(1|
|00006220| 2d 66 27 28 73 29 5c 62 | 69 67 72 29 7d 2e 24 24 |-f'(s)\b|igr)}.$$|
|00006230| 0d 0d 53 75 62 73 74 69 | 74 75 74 69 6e 67 20 74 |..Substi|tuting t|
|00006240| 68 69 73 20 76 61 6c 75 | 65 20 6f 66 20 24 74 24 |his valu|e of $t$|
|00006250| 20 69 6e 74 6f 20 74 68 | 65 20 66 6f 72 6d 75 6c | into th|e formul|
|00006260| 61 20 66 6f 72 20 24 5c | 66 5f 74 28 73 29 24 0d |a for $\|f_t(s)$.|
|00006270| 74 65 6c 6c 20 75 73 20 | 74 68 61 74 20 74 68 65 |tell us |that the|
|00006280| 20 63 75 73 70 20 6d 75 | 73 74 20 62 65 20 74 68 | cusp mu|st be th|
|00006290| 65 20 70 6f 69 6e 74 0d | 24 24 0d 5c 6c 65 66 74 |e point.|$$.\left|
|000062a0| 28 7b 31 5c 6f 76 65 72 | 31 2d 66 27 28 73 29 7d |({1\over|1-f'(s)}|
|000062b0| 2c 5c 20 73 2b 7b 66 28 | 73 29 2d 73 5c 6f 76 65 |,\ s+{f(|s)-s\ove|
|000062c0| 72 31 2d 66 27 28 73 29 | 7d 5c 72 69 67 68 74 29 |r1-f'(s)|}\right)|
|000062d0| 0d 3d 5c 6c 65 66 74 28 | 7b 31 5c 6f 76 65 72 31 |.=\left(|{1\over1|
|000062e0| 2d 66 27 28 73 29 7d 2c | 5c 20 7b 66 28 73 29 2d |-f'(s)},|\ {f(s)-|
|000062f0| 73 66 27 28 73 29 5c 6f | 76 65 72 31 2d 66 27 28 |sf'(s)\o|ver1-f'(|
|00006300| 73 29 7d 5c 72 69 67 68 | 74 29 2e 24 24 0d 0d 41 |s)}\righ|t).$$..A|
|00006310| 70 70 6c 79 69 6e 67 20 | 74 68 65 20 64 65 73 63 |pplying |the desc|
|00006320| 72 69 70 74 69 6f 6e 20 | 6f 66 20 74 68 65 20 65 |ription |of the e|
|00006330| 6e 76 65 6c 6f 70 65 20 | 67 69 76 65 6e 20 62 79 |nvelope |given by|
|00006340| 20 50 72 6f 70 6f 73 69 | 74 69 6f 6e 0d 31 20 6e | Proposi|tion.1 n|
|00006350| 6f 77 20 63 6f 6d 70 6c | 65 74 65 73 20 74 68 65 |ow compl|etes the|
|00006360| 20 70 72 6f 6f 66 2e 0d | 5c 62 69 67 73 6b 69 70 | proof..|\bigskip|
|00006370| 0d 0d 5c 62 65 67 69 6e | 73 65 63 74 69 6f 6e 20 |..\begin|section |
|00006380| 7b 43 6f 6e 63 6c 75 73 | 69 6f 6e 73 7d 0d 0d 54 |{Conclus|ions}..T|
|00006390| 68 65 20 44 79 6e 61 6d | 69 63 20 46 75 6e 63 74 |he Dynam|ic Funct|
|000063a0| 69 6f 6e 20 56 69 73 75 | 61 6c 69 7a 65 72 20 70 |ion Visu|alizer p|
|000063b0| 72 65 73 65 6e 74 73 20 | 6e 65 77 20 61 6e 64 20 |resents |new and |
|000063c0| 69 6c 6c 75 6d 69 6e 61 | 74 69 6e 67 0d 70 69 63 |illumina|ting.pic|
|000063d0| 74 75 72 65 73 20 6f 66 | 20 66 75 6e 63 74 69 6f |tures of| functio|
|000063e0| 6e 73 20 63 6f 6e 73 69 | 64 65 72 65 64 20 6e 6f |ns consi|dered no|
|000063f0| 74 20 61 73 20 73 74 61 | 74 69 63 0d 67 72 61 70 |t as sta|tic.grap|
+--------+-------------------------+-------------------------+--------+--------+